- 1 - Appeared in: Journal of Structural Engineering, Trans. ASCE, 2011, 137(12): 1573–1582. Reliability Acceptance Criteria for Deteriorating Elements of Structural Systems Daniel Straub 1 & Armen Der Kiureghian 2 Abstract A systematic approach to determining reliability-based acceptance criteria for deteriorating elements in structural systems is proposed, as a basis for calibration of safety factors in codes and standards and for verifying acceptability of inspection and maintenance strategies for specific structures. The goal is to establish deterioration acceptance criteria for the elements of a structural system in compliance with criteria formulated for the system. Existing methods significantly overestimate the deterioration reliability of redundant structural systems because they neglect the joint effect of deterioration failures of different elements. To more realistically capture the load-sharing behavior of deteriorating redundant structural systems, it is proposed to establish deterioration acceptance criteria based on easily computable, idealized structural systems, which are calibrated to the characteristics of the real structure. The approach is validated on an example structural system and is found to represent a significant improvement over current methods. The paper concludes with a study of the main factors influencing acceptance criteria of deterioration reliability. 1 Associate Professor, Engineering Risk Analysis Group, Technical University Munich, Arcisstr. 21, 80290 München, Germany. Email: [email protected]2 Taisei Professor of Civil Engineering, Dept. of Civil & Environmental Engineering, Univ. of California, Berkeley, CA 94720. Email: [email protected]
Transcript
- 1 -
Appeared in: Journal of Structural Engineering, Trans. ASCE, 2011, 137(12): 1573–1582.
Reliability Acceptance Criteria for Deteriorating Elements of Structural
Systems
Daniel Straub1 & Armen Der Kiureghian2
Abstract
A systematic approach to determining reliability-based acceptance criteria for deteriorating
elements in structural systems is proposed, as a basis for calibration of safety factors in codes
and standards and for verifying acceptability of inspection and maintenance strategies for
specific structures. The goal is to establish deterioration acceptance criteria for the elements of
a structural system in compliance with criteria formulated for the system. Existing methods
significantly overestimate the deterioration reliability of redundant structural systems because
they neglect the joint effect of deterioration failures of different elements. To more realistically
capture the load-sharing behavior of deteriorating redundant structural systems, it is proposed
to establish deterioration acceptance criteria based on easily computable, idealized structural
systems, which are calibrated to the characteristics of the real structure. The approach is
validated on an example structural system and is found to represent a significant improvement
over current methods. The paper concludes with a study of the main factors influencing
acceptance criteria of deterioration reliability.
1 Associate Professor, Engineering Risk Analysis Group, Technical University Munich, Arcisstr. 21, 80290 München, Germany. Email: [email protected] 2 Taisei Professor of Civil Engineering, Dept. of Civil & Environmental Engineering, Univ. of California, Berkeley, CA 94720. Email: [email protected]
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Introduction
Owners of structural systems are confronted with the problem of determining whether their
structures and their inspection/maintenance/repair policies are acceptable with regard to
potential deterioration failures. While this applies equally to newly built and existing
structures, the problem is particularly relevant for the latter, for which the cost of increasing the
reliability is generally much higher (Melchers 2001). Extensive research has been carried out
on probabilistic modeling of deterioration in structural systems, as reviewed by Frangopol et al.
(2004). Furthermore, methods for reliability analysis of deteriorating structural systems have
been developed over the past two decades, including works by Mori and Ellingwood (1993), Li
(1995), Ciampoli (1998), Estes and Frangopol (1999) and Stewart and Val (1999). These
methods enable the computation of the time-variant reliability of structures with deteriorating
elements, which in the general case is a highly complex task. Because of this complexity, such
integrated reliability analysis is rarely performed in engineering practice; rather, deterioration
is assessed at the level of structural details or elements. The present paper, therefore, follows a
different strategy. It proposes a method for determining the required level of deterioration
reliability at the level of structural elements that ensures acceptability of the risks at the
structural system level. The method accounts for the relevant influencing factors in an
approximate sense, while remaining sufficiently simple for practical applications. The approach
is motivated by practices in structural engineering for fixed offshore structures applied since
the 1980s, where acceptance criteria for fatigue deterioration are determined as a function of
the structural redundancy with respect to element failure (Kirkemo 1990, Moan 2005). The
proposed method can be applied to determine acceptability of specific structures and
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inspection/maintenance strategies (Straub and Faber 2005b), or for calibration of safety factors
for deterioration limit states in codes and standards.
Problem setting
Existing codes typically specify design criteria and safety factors for individual structural
elements. This applies for failures caused by static or dynamic overloading of the structural
elements, as described by ultimate limit states, as well as for deterioration failures, e.g.,
described by fatigue limit states. However, deterioration failures exhibit some fundamental
differences as compared to overloading failures, which make it necessary to explicitly account
for the system characteristics. When structural systems collapse because of overloading, all
elements involved in the realized failure mode normally fail during the same load event (when
considering cascading failure sequences as a single event). For this reason, the safety margins
of the individual elements exhibit strong statistical dependence and the system reliability
approximately equals the reliability of the individual elements (assuming that all elements have
been designed to have the same target reliability index). Failures of structural elements caused
by deterioration, on the other hand, are likely to occur at different times depending on the
nature of the deterioration process. These events have lower statistical dependence and the
corresponding system reliability, therefore, substantially differs from the deterioration
reliability of the individual elements. For these reasons, the acceptability of deterioration
failures must be assessed as a function of structural redundancy. In addition, deterioration can
be detected before failure occurs, but deterioration failures can also remain undetected. The
inspection and repair policies, therefore, influence the acceptability of deterioration failures.
These aspects are partly reflected in design codes such as Eurocode 3 (1992) and NORSOK
(1998), where safety factors for fatigue limit states are specified as a function of the
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consequences of element failure (structural importance) and the possibility to inspect an
element.
In the past, reliability-based acceptance criteria for deterioration limit states have been
considered mainly for structures subject to fatigue, in particular fixed offshore structures, e.g.,
HSE (2002), Ronalds et al. (2003), Moan (2005), and Straub and Faber (2005a). These
acceptance criteria were determined as a function of the structural importance of the considered
element. The structural importance of each element was assessed by comparing the overall
capacity of the intact structural system with the capacity of the structural system when the
element is removed. As shown in this paper, this approach is only suitable for elements with
high reliability and if the statistical dependence among deterioration failures is low, because it
neglects the possibility of joint occurrence of more than one deterioration failure. These
conditions often are not satisfied in practice. For many structural systems, deterioration states
of structural elements are correlated. As previously stated, this correlation is lower than that for
limit states of element failures due to overloading (for which the correlation coefficient is close
to one), but it is non-negligible for most structural systems. As an example, in an investigation
of the integrity of mooring systems for floating offshore structures, it was found that
deterioration typically affects all mooring lines to roughly the same extent (HSE 2006), which
implies large statistical dependence among the corresponding deterioration states. Since
mooring systems have significant redundancy (failure of an individual mooring line is
generally not critical), the dependence among the deterioration processes of different elements
strongly influences the system reliability. Another example is the study reported by
Vrouwenvelder (2004), which inferred statistical dependence among fatigue performance of
welded joints by comparing within-batch variability to batch-to-batch variability. On this basis,
the correlation coefficient between fatigue crack growth parameters of two welded joints in the
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same structure was estimated as 0.85. Thus, fatigue failure events in structural systems are
expected to exhibit significant statistical dependence, which must not be neglected.
The goal of this paper is to introduce a practical yet scientifically sound method for
determining reliability acceptance criteria for deteriorating elements in general structural
systems, as a function of overall system acceptance criteria, the structural importance of the
individual element within the system, the statistical dependence among deterioration failures
throughout the structure and the inspection and repair policy. Let TDS denote the target
reliability index associated with the deteriorated structural system and TDEi denote the target
reliability index for deterioration failure of the ith element in the structure. Our objective is to
determine TDEi so that the overall system complies with T
DS . This approach is motivated by
the Probabilistic Model Code of the Joint Committee on Structural Safety, JCSS (2006), where
target reliabilities are specified for the entire structural system based on socio-economical
principles, including life safety aspects.
Target reliability indices for the structural system
In the Probabilistic Model Code of the Joint Committee on Structural Safety (JCSS 2006),
target reliability indices T for ultimate limit states are specified as a function of the
consequences of component failure and the relative cost of a safety measure, see Table 1. This
differentiation reflects the fact that the target reliability indices are based on an optimization of
expected life-cycle costs (Rackwitz 2000). The values in Table 1 are equally valid for new and
existing structures, but the relative cost of safety measures is typically larger for the latter,
leading to generally lower target reliability indexes for existing structures.
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Table 1. Tentative target reliability indices T for ultimate limit states and one year reference period, as
recommended in JCSS (2006).
Relative cost of safety measure
Minor consequences of failure
Moderate consequences of failure
Large consequences of failure
Large T=3.1 (pF T
10-3) T=3.3 (pF T
5 10-3) T=3.7 (pF T
10-4)
Normal T=3.7 (pF T
10-4) T=4.2 (pF T
10-5) T=4.4 (pF T
5 10-6)
Small T=4.2 (pF T
10-5) T=4.4 (pF T
5 10-6) T=4.7 (pF T
10-6)
According to JCSS (2006), “the values given [in Table 1] relate to the structural system or in
approximation to the dominant failure mode.” In the absence of owner-specified reliability
targets, these values can be considered as the target reliability indices associated with
deterioration-induced system collapse, TDS . Mitigation measures against deterioration typically
are expensive, in particular for existing structures, and the target reliability indices in most
cases will be as given in the upper two lines of Table 1.
System model
To verify compliance with TDSβ , a model for computing the probability of collapse of the
deteriorating structural system, DSp , and the corresponding reliability index DS , is needed.
Deterioration in a system is deemed acceptable if DSTDS .We propose a formulation based
on a simplified model of the element and system behavior. The first simplification is that, on a
system level, deterioration of any element i at time t is modeled by a binary random process
( )iE t with outcome space { , }i iF F , iF being the event of deterioration failure of the element
and a superposed bar indicating the complement. Thus, no gradual decay of the element
strength is considered: At a given time t , the element either has its full capacity (not
deteriorated) or has completely lost its capacity due to deterioration. (The appropriateness of
this idealization is discussed later in the section on deterioration models.) The deterioration
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state of the system, represented by the random process ( )t , is a function of ( )iE t ,
1,2,...,i n , where n is the number of deteriorating structural elements. The outcome space of
( )t thus consists of 2n disjoint states, i , ni 2,,1 . The first of these states corresponds to
the event of no deterioration failure in the structural system, 1 1 2{ ... }nF F F , and the
last to the event that all elements have failed due to deterioration, }{ψ 212 nFFFn .
A second simplification is that the deterioration state of the system is constant over a time
period 1t year, which is considered to be small in relation to the service life of the
structure. To be on the conservative side, the system deterioration state in the period ],( ttt
is set equal to the state at time t , ( )t . The event of structural collapse in that time interval is
denoted by )(tC . The probability of this event conditioned on the deterioration state of the
system at time t , )](|)(Pr[ ttC , can be computed by performing reliability analyses of the
structure with the elements damaged according to (t), i.e., all elements that are failed due to
deterioration are removed in the structural model employed in the reliability analysis. However,
for real structures it is not feasible to evaluate all 2n values of )](|)(Pr[ ttC , as this would
require an enormously large number of system reliability analyses ( 2n being the size of the
outcome space of (t)). To circumvent this problem, later in this paper we propose to compute
)](|)(Pr[ ttC for an approximately equivalent idealized system, which is constructed based on
a set of indicators of the real structural system.
The probability of structural collapse in the reference period ],( ttt is given by the total
probability theorem as
ii
iDS tttCtCtpn
PrPrPr2
1
(1)
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The associated reliability index is 1( ) [ ( )]DS DSt p t , where 1 is the inverse of the
standard normal cumulative distribution function. Pr[ ( ) ]it in Equation (1) is obtained as
a function of probabilities of the element deterioration failure events ii FtE )( , accounting for
the statistical dependence among these events. Because we can set )β(])(Pr[ TDEiii FtE
for an element designed at the limit of the acceptance criterion, Equation (1) establishes the
connection between the system criterion DSTDS and the target deterioration reliability
indices of the individual elements, TDEi , 1,2,...,i n . Obviously, the single condition
DSTDS is not sufficient to determine the n individual quantities T
DEi and additional rules are
required. Such rules are proposed in this paper, based on the same equivalent idealized system
as introduced for computing Pr[ ( ) | ( ) ]iC t t .
Modeling deterioration failure events
Deterioration is modeled at the level of structural elements, e.g., structural members, welded
joints, area segments of a continuous surface. The event of deterioration failure of element i at
time t is represented by a limit-state function ),( tgi X , with X being a vector of random
variables that describe the deterioration model, so that }0),({})({ tgFtE iii X . The
corresponding failure probability, Pr[ ( ) ]i iE t F , can be computed by the methods of structural
reliability analysis. An example deterioration limit state model is
BAtDtg ,X (2)
where t is the time since installation or repair of the element, D is the damage limit and A
and B are parameters describing the deterioration process. For 1B , this corresponds to most
applied corrosion models as well as to the Palmgren-Miner fatigue model with a stationary
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stress process; for 5.0B , the model is representative of diffusion-controlled deterioration,
and for 2B the model approximates concrete deterioration due to sulfate attack.
Deterioration in an element occurs gradually with time and representing the capacity of such an
element by the two-state random process ( )iE t is a strong simplification. Therefore, care is
required in defining the failure criterion in the deterioration limit state function, such as D in
Equation (2). If the failure event is defined so that the capacity of the element is significantly
reduced before the limit state is reached, the binary model can be unconservative. On the other
hand, if the failure event is defined so that the element is considered failed after a small loss of
capacity, the model will give conservative results for the system. In general, the assumed
binary model would be most appropriate when the deterioration initiates and failure occurs
within the same time interval ],( ttt .
For fatigue deterioration, limit states provided in codes generally correspond to defect initiation
or the event of a through-thickness crack and not to loss of capacity; the remaining capacity of
the element or joint at the limit state may be close to its capacity in the undamaged state.
Therefore, the proposed model is conservative for fatigue limit states; however, the degree of
conservatism can vary. For some structural details, fatigue can lead to unstable crack growth
and complete loss of capacity shortly after reaching the limit state and the model is accurate.
On the other hand, in many structural configurations loads redistribute once a loss of stiffness
occurs and crack growth slows down after the limit state is reached; the model is conservative
in this case. Despite its potential conservatism, we believe the proposed binary model is
justified for modeling high-cycle fatigue failures in engineering practice. For low-cycle fatigue,
however, the model can be non-conservative. Damaging stress cycles due to low-cycle fatigue
usually occur during extreme events, and it is more probable that deterioration failures and
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structural collapse occur during the same load event. By not accounting for this likely
concurrence, the model might underestimate the probability of collapse.
The binary model is suitable for other deterioration processes that lead to rapid reduction of
capacity after an initiation period. These include various forms of stress corrosion cracking and
deterioration processes that are controlled by a protection system. In the latter case, the
deterioration failure event should be defined (conservatively) as the failure of the protection
system.
For other deterioration mechanisms that lead to slow reduction of the element capacity, such as
uniform corrosion or distributed pitting corrosion on steel surfaces and on reinforcement of RC
structures, the binary model is less appropriate. It might still be applied if the failure criterion is
selected conservatively, e.g., by defining the allowable corrosion loss in ship structures as the
damage limit in Equation (2) or by defining the failure of the reinforcement as corrosion-
induced loss of bond. Depending on the application, the results obtained with the model
presented in this paper can be overly conservative and approaches based on structure-specific
system reliability analyses might become necessary. However, it is noted that for deterioration
of RC structures, serviceability limit states are often found to be determining the required level
of deterioration reliability (Stewart and Val 2003). In this case, the present approach can still be
used to check whether the reliability levels implied by the serviceability criteria are complying
with the system safety criterion.
Modeling statistical dependence among deterioration failure events
The deterioration failure events of elements in a structural system are generally statistically
dependent due to common uncertain influencing factors, such as environmental conditions and
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material characteristics. Statistical dependence among element deterioration failures can be
expressed through the correlation coefficients among the corresponding limit state functions.
As an example, consider the deterioration limit state in Equation (2). This can be reformulated
into the equivalent form
tBADtg lnlnln, X (3)
If both D and A are modeled by a Lognormal distribution and B is modeled by a Normal
distribution, assuming independence of the three variables, the reliability index at time t
without inspection becomes
222 lnσζζ
lnμλλβ
t
tt
BAD
BADDEi
(4)
with Dλ , A and Bμ being the means of ln , Aln and B , and , A and Bσ being the
corresponding standard deviations, respectively. As an example, assume the statistical
dependence between the deterioration failures of two elements i and j arises due to
correlation between the corresponding variables iDln and jDln and between iAln and jAln ,
denoted ln A and ln , respectively, while variable B remains statistically independent from
element to element. Assuming identical marginal probability distributions of these variables for
the two elements, the correlation coefficient between the corresponding pairs of limit states
functions is
2 2ln ln
22 2 lnA A
M
A B
tt
(5)
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For the special case considered here, with the limit-state functions being jointly normally
distributed, the pair-wise correlation coefficients M t together with )(tDEi fully describe the
probability mass function (PMF) of )(t , i.e., the probabilities of all possible combinations of
element deterioration failures in the system. In the more general case, when the deterioration
limit state function is not linear and the random variables are not normal or lognormal, M t
can be taken as the correlation coefficient between the linearized limit states obtained from a
FORM solution of a parallel system with two elements (Ditlevsen and Madsen 1996).
Investigation of earlier models for developing deterioration acceptance criteria in redundant structural systems
In principle, to establish the element acceptance criterion TDEi as a function of the system
acceptance criterion TDS , it is required to solve Equation (1). Because of the difficulty in
computing ])()(Pr[ ttC for all outcomes of )(t , existing approaches (HSE 2002, Ronalds et
al. 2003, Moan 2005, Straub and Faber 2005a) employ an approximate version of Equation
(1). As an example, HSE (2002) utilizes the following approximation:
ii
n
iniiiDS FtEFFFFFttCttCtp
Pr......PrPr
11111
(6)
Here, the influence of individual deterioration failures iF is appraised through the probability
of system failure with element i removed and all other elements
intact: 1 1 1Pr[ ( ) | ( ) ( )]i i i nC t t F F F F F . This conditional probability has
often been used as an indicator for redundancy of the structure with respect to failure of
element i (Lotsberg and Kirkemo 1989, Gharaibeh et al. 2002). The approach based on Eq. (6)
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requires only one additional reliability analysis per element, i.e., n analyses instead of 2n ,
which makes it practically feasible. By comparing Equations (1) and (6), it can be seen that the
two formulations are identical if the element deterioration failure events are mutually exclusive
and if the probability of collapse of the intact structure is zero, 0)()(Pr 1 ttC . As
discussed in Straub and Faber (2005a), and as demonstrated by a numerical example later in
this paper, the approximation is reasonable when the individual structural elements have high
deterioration reliability ( 5.3)( tDEi ), when the number of structural elements is small and
when deterioration failure events are uncorrelated. In such a case, the probability of the joint
occurrence of two or more deterioration failures becomes negligible. (If all elements have the
same failure probability pFtE ii ])(Pr[ , the probability of more than one statistically
independent failure event among n elements is 2/)1()1()1(1 21 pnnpnpp nn , which
is much smaller than p when p is small and n is of order smaller than p/1 .)
Unfortunately, these conditions are not generally fulfilled for real structures.
Motivated by the approximation in Equation (6), we define the Single-Element Importance
(SEI) measure for element i as
1 1 1 1 1 1Pr ( ) | ( ) ... ... Pr ( ) | ( )i i nSEI C t t F F F F F C t t (7)
As can be seen, iSEI is the difference in the failure probability of the system with all elements
intact (not deteriorated) and the system where only element i has failed due to deterioration.
In addition to Equation (6), further conditions are required to establish the element acceptance
criteria. It has been suggested, explicitly in (Straub and Faber 2005a) and implicitly in (Ronalds
et al. 2003, Moan 2005, HSE 2002), to determine the TDEi such that all summation terms in
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Equation (6) are equal, i.e., all elements contribute equally to the probability of system failure
associated with deterioration. The target reliability indices for all elements are then obtained as:
1
11
Pr
Pr1
ttCSEI
ttC
n i
TDST
DEi (8)
Both Equations (6) and (8) neglect the contribution of joint deterioration failure events of two
or more elements. To examine this effect, in the following an idealized system, for which
Pr[ ( ) | ( )]C t t is easily computable, is investigated.
To simplify the notation, hereafter )(β tDEi is written as DEi , because the structure is verified
under the assumption that the element deterioration reliability is at its limit, i.e. TDEiDEi t β)(β ,
which does not depend on time. In addition, the random variables ( )C t and ( )t are written as
C and , since the probability Pr[ | ]C does not change with time under the common
assumption that the distribution of the annual maximum load is constant with time and )Pr(
does not change with time if the DEi are constant with time.
The SEI for a Daniels system
Consider the Daniels system (Daniels 1945) shown in Figure 1. The elements of the system
have independent and identically distributed (iid) capacities, i.e. they are exchangeable in the
statistical sense. In Gollwitzer and Rackwitz (1990), the characteristics of this system are
examined for a variety of element behaviors. This idealized system is well suited for
representing the load-sharing phenomenon present in structural systems, with the two cases (a)
and (b) in Figure 1 representing the extremes of true material behavior. Note that the distinction
between the brittle and ductile failure modes relates to element failures due to overloading of
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the structure. Deterioration, on the other hand, affects the capacities of the elements. In the
simplified model considered here, the deterioration failure of an element is tantamount to
reduction of its capacity to zero. The deterioration state of the system essentially dictates the
number of elements that are available to resist the applied load through either a ductile or brittle
behavior.
For the idealized system, computation of the SEIi according to Equation (7) is straightforward.
The two needed terms are
1 1 1 1 1Pr | ... ... Pr | 1i n FC F F F F F C N (9)
1Pr Pr 0FC C N (10)
where FN is the number of elements failed due to deterioration.
EI =
R1 R2 Rn
Lε
RRi
. . .R3 ε
RRi
Case a)
Case b)
∞
Figure 1. Idealized structural system under external load. Case a) brittle element behavior (original Daniels system), b) ductile element behavior.
To evaluate Equations (9) and (10), the conditional failure probability Pr( | )FC N j is
required. For given probability distributions of the element capacities iR and the load L , this
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is readily obtained for the above system. In accordance with the definition of C , L is the
maximum load in the period ],( ttt . For case a), the required conditional probability is
calculated as
Pr Pr ,F L
L
C N j C l n j f l dl (11)
where the probability of system failure for given load l and number of surviving elements
)( jn , Pr[ | , ( )]C l n j , is computed according to the solution provided in Daniels (1945). For
case b), the solution is given by
1
Pr Pr 0n j
F ii
C N j R L
(12)
which is easily computed using structural reliability methods.
Because of exchangeability of its elements, Equation (1) for the Daniels system simplifies to
n
iFFDS jNjNCp
0
)Pr()|Pr( (13)
The probability that j elements have failed due to deterioration, Pr[ ]FN j , is a function of
the element deterioration reliability indices DEi and the correlation coefficients M between
their limit states. We assume DEi is the same for all elements and M is the same for all pairs
of elements. The probability of j deterioration failures among N elements then is
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Pr
with 1
n j j
F
DEi M
M
nN j u p u p u du
n j
up u
(14)
where ( ) is the standard Normal probability density function. This equation is based on a
binomial model with uncertain parameter p, which accounts for the statistical dependence
among the Bernoulli trials according to the correlation coefficient M .
Numerical investigations
With the Daniels system as an example of a structural system, we can now investigate the
effect of the approximation made in existing approaches for determining the deterioration
target reliability index. This is done by comparing the true deterioration reliability of the
Daniels system with the one computed according to Equation (6). For this purpose, the load, L ,
is modeled by a lognormal distribution with coefficient of variation (c.o.v.) 0.25L and the
capacities of the elements, iR , are modeled by independent and identical normal distributions
with c.o.v. 15.0δ R . The ratio of the mean values of inR and L , which can be considered as
the mean safety factor for system overload failures, is determined such that the system in its
undamaged state (without deterioration failures) has reliability index
11{Pr[ | ]} 4.4
DSC . (This value has reference period 1yrt , but is not
dependent on time t .) For a system with 20n elements, this gives / 3.67iR Ln for the
brittle material behavior and / 2.90iR Ln for the ductile behavior. For this system,
Pr( | )FC N j is illustrated in Figure 2 as a function of j for the two material models as
computed by use of Equation (14). It is observed that the criticality of deterioration failures is
almost identical for the two material behaviors. (It is reminded that the difference in material
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behaviors relates only to overload failures. Deterioration failures for both material behaviors
are modeled as brittle, i.e., without remaining load capacity.) In the remainder of this section,
only the system with ductile elements is considered.
10 15 20
10-5
10-4
10-3
10-2
10-1
1.0
Number of elements failed due to deterioration, j0
Cond
ition
al p
roba
bilit
y of
col
laps
e Pr
(C |
NF =
j )
5
Brittle material (original Daniels system)Ductile material
Figure 2. Failure probability of Daniels system as a function of the number of elements failed due to deterioration.
To appraise the effect of the approximation introduced in previous approaches to determining
the deterioration acceptance criteria, we compute the system reliability associated with
deterioration failures, 1( )DS DSp , according to Equation (6) and Equation (13). Equation
(6) represents the approximation used in previous approaches and is based on the SEIi, which
here is the same for all elements and is computed as )0|Pr()1|Pr( FFi NCNCSEI .
Equation (13) gives the exact value of DS for the Daniels system and is used as a reference. In
Figure 3, DS is shown as a function of the number of elements, n, the deterioration reliability
index of the individual elements, iDE , and the pair-wise correlation coefficient among the
deterioration safety margins, M .
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0 5 10 15 20 25 303.0
3.5
4.0
4.5
Number of elements
Syst
em re
liabi
lity
inde
x β D
S
Syst
em re
liabi
lity
inde
x β D
S
1.0 1.5 2.0 2.5 3.0 3.5 4.01.5
2.5
3.0
3.5
4.0
4.5
Element deterioration reliability index βDEi
0 0.2 0.4 0.6 0.8 1.03.0
3.5
4.0
4.5
Pair-wise correlation coefficient among deterioration safety margins, ρM
Syst
em re
liabi
lity
inde
x β D
S
2.0ρM = 0.4
βDEi = 3.0
n = 20
ρM = 0.4
Approximation (Eq. 6)
Correct result (Eq. 13)
n = 20
βDEi = 3.0
Figure 3. System deterioration reliability index as a function of number of elements (left chart), element deterioration reliability index (middle chart), and correlation among deterioration limit states (right
chart).
The results in Figure 3 clearly demonstrate that the approximation made in previous approaches
to determining deterioration acceptance criteria overestimates the reliability of the investigated
system, and the same tendency is expected for every redundant structural system. This effect is
relatively constant with the number of elements in the Daniels system, n , except when n is
close to one, representing systems with limited or no redundancy. Furthermore, as mentioned
earlier, the approximation is close to the correct result when the deterioration reliability index
of the individual elements is large and when the statistical dependence among deterioration
failure events is low ( 3.0M ). In these cases, the probability of joint occurrence of several
element deterioration failures is negligible. However, for most real structural systems, these
assumptions do not hold, and an improved approximation to the actual system deterioration
reliability DS is required. Such an approximation is presented and investigated in the
remainder of this paper.
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Acceptance criteria for deteriorating structural elements in general redundant systems
Equivalent structural systems
Our aim here is to set a target reliability index TDEiβ for each deteriorating element of a
structural system so that the system reliability index considering deterioration, DSβ , is no less
than a specified target reliability index TDSβ . Obviously T
DEiβ may need to be different for
different elements, depending on the relative structural importance of each element. The
relationship between element and system reliability indices, however, is an intricate one,
governed by the nature of load-sharing between the elements, the configuration of the system
and, in particular, the distribution of deteriorating elements within the structure. It is
impractical to use an exact representation of the system (e.g., as a series system of parallel
subsystems, Hohenbichler and Rackwitz 1982) to establish this relationship. Instead, here we
make use of an idealized “equivalent” representation of the system to determine the required
relationship. It is desirable to choose an idealized system with exchangeable (statistically
independent and identically distributed) elements, because this property facilitates computation
of the relation between TDEiβ and DSβ , as earlier demonstrated for the Daniels system..
However, the elements in the real structure have varying importance and cannot be represented
as exchangeable elements within a single idealized system. Therefore, a different idealized
system is defined for each deteriorating element in the real structure.
For the idealized system to provide an accurate representation, it must be calibrated to the
reliability characteristics of the real element and the real structure. Hence, for each element, the
corresponding idealized system is defined so that it correctly represents the reliability of the
intact structure and the reliability of the structure with the element removed. The difference
between these two reliability measures, which is equal to the SEI of the element, in a sense
- 21 -
reflects the redundancy of the real system with respect to the selected element. Additionally,
the idealized system should reflect the total number of deteriorating elements in the real
structure, n. This is because, for given reliability of the intact structure and its redundancy with
respect to the selected element, a larger n implies a higher number of failure modes and
consequently lower system reliability. To assure satisfaction of the overall system reliability
requirements, the target reliability index for the selected element must account for n.
For each element i in the real structure, the proposed equivalent idealized system consists of a
set of k Daniels subsystems in series, each having in elements with statistically independent
and identically distributed capacities. in is selected so that it represents the redundancy of the
real structural system with respect to deterioration failure of element i; when this redundancy is
large, equivalent Daniels subsystems with larger number of elements are used, wherein failure
of one element has a smaller effect. Since in is determined purely based on the redundancy of
the system with respect to element i , it does not reflect the total number of elements in the real
system. For this reason, k subsystems are considered in series, where k is selected to
appropriately represent the total number of elements in the real structure n . A larger value of
n for constant in implies a larger value of k . The numerical determination of in and k is
described later.
The deterioration failure events of the elements within each Daniels system with in elements
are characterized by the common target reliability index TDEiβ and the common correlation
coefficient M , which represents the dependence of the deterioration failure of element i on
those of other elements, e.g. computed according to Eq. (5). Deterioration failure events in
different Daniels subsystems are assumed to be statistically independent. The loads acting on
the k subsystems are statistically independent and identically distributed. Due to this latter
- 22 -
assumption, which is necessary to maintain exchangeability of the elements, the system cannot
be interpreted as a single structural system. Instead, it is a logical system, which fails if any of
its k Daniels subsystems fails. The idealized system is illustrated in Figure 4.
Figure 4. The equivalent system for element i. In their undeteriorated state, all ni·k elements have independent and identically distributed capacities jiR , relative to overload failure. System failure
occurs if any of the k subsystems fails.
The distributions of the loads jL and the element capacities ijR , must be selected so as to
represent the characteristics of the dominant load case, and the parameters are selected so that
the idealized system in its intact state has the same reliability as the real structure without
deterioration failures. As an example, for an offshore structure in a hurricane-prone area,
typical values of the c.o.v. are 35.0L and 15.0iR (Stahl et al. 2000). These values are
utilized in the numerical examples in this paper, and it is assumed that jL is modeled by a
Lognormal distribution and ijR , by a Normal distribution. The ratio between the mean values
of jL and ,i j in R is determined by matching the reliabilities of the real and idealized systems in
their intact (not deteriorated) states. Specifically, the ratio /ii R Ln is determined iteratively
from the condition
1
1Pr 0 1 ki F DS
C N (15)
- 23 -
where DS
is the reliability index of the real structure in its intact state, iC is the event of
failure of a Daniels system with in elements and Pr( | 0)i FC N is computed according to
Equation (11) or (12).
in , the number of elements in each Daniels system, represents the redundancy of the real
structural system with respect to deterioration failure of element i. Specifically, in is selected
as the number of elements of the Daniels system for which the (exchangeable) elements have
the same SEI as element i in the real structure. The SEI of the elements in the equivalent
system, denoted by iSEI , is obtained as
11 1 Pr 0 1 Pr 1
k
i F F DSSEI C N C N
(16)
Here, Pr( )FC N j is the probability of failure of a Daniels system with in elements, of
which j elements have failed due to deterioration, and is given by Equations (11) and (12).
Since in is an integer variable, the iSEI computed for element i of the real structure cannot be
exactly matched. Instead, the two integer values of in that give iSEI values closest to iSEI are
determined and the analysis is carried out for the two systems.
Since in is not a direct function of the number of deteriorating elements in the real structure n ,
the effect of n on the system reliability is accounted for by k , the number of Daniels systems
in series. For given values nini ,...,1, , k can be determined as the sum of the contributions of
the elements in their respective equivalent systems, which can be stated as
1
1n
i i
kn
(17)
- 24 -
Alternatively, k can be determined from the condition that the mean number of elements in the
equivalent subsystems should be equal to the true number of deteriorating elements. It then
follows that
n
iin
nk
1
2
(18)
Hereafter, we employ Equation (17), but we note that Equation (18) gives similar results and
both formulations give exact results in the extremes: for a series system with n elements where
1in for all elements, both equations correctly give nk , and for a parallel system with n
elements where nni , they correctly give 1k .
So far we have described how the parameters defining the equivalent systems, i.e., k ,
nini ,...,1, and the ratio between the mean values of jL and ,i j in R , are obtained separately,
assuming that the other parameters are given. To determine all parameters jointly, an iterative
procedure is utilized. An initial guess of k is made, and the remaining parameters are
determined for the given k . With the resulting values of nini ,...,1, , a new value of k is
computed and the process is repeated until convergence in k is achieved. Figure 5 summarizes
the procedure for determining the parameters of the equivalent systems. The computational
effort for this procedure is reasonable and not critical for practical implementations (in the
order of seconds on a standard Pentium II PC for an implementation in Matlab).
- 25 -
Algorithm for establishing the equivalent systems Input: System target deterioration reliability T
DSβ ; reliability of the system without deterioration SDβ ; structural importance of all deteriorating elements niSEIi ...1, = ; distribution model and the c.o.v. of Lj and ijR , ; material behavior (brittle, ductile). Output: Parameters k and nini ...1, = describing the equivalent systems. 1. Make an initial guess of k : 0kk ←′ 2. Select an initial range for the equivalent numbers of elements
i,maxi,mini nn :←′n 3. For all jn in in′ do:
⋅ determine ( )/jj R L tn μ μ from the condition in Equation 18;
⋅ determine j js SEI′ ′= as a function of jn according to Equation 21.
4. If not )max()min( ss ′≤≤′ iSEI for all ni ...1= , then: ⋅ if not iSEI≤′)min(s for all ni ...1= , then select a new,
higher value for i,maxn ⋅ if not )max(s′≤iSEI for all ni ...1= , then select a new,
lower value for i,minn ⋅ i,maxi,mini nn :←′n ⋅ go to 3.
5. For all ni ...1= , determine in as a function of iSEI by interpolation from in′ and s′ .
6. )/1(1 ini nk =Σ←
7. If not tolkktolk +′≤≤−′ then kk ←′ ; go to 2. Else end.
Figure 5. Algorithm for establishing the equivalent systems.
Determination of the element acceptance criterion from the equivalent system
Once the equivalent system for element i is established, this system is utilized to determine the
element deterioration acceptance criterion TDEi . The equivalent system has exchangeable
elements, so all its elements have the same deterioration reliability index. The value of TDEi is
determined from the condition TDSDS , where DSβ is the reliability index associated with
deterioration failures in the equivalent system.
The probability of failure of the equivalent system is
k
n
jFFDS
i
jNjNCCp
0
PrPr11Pr (19)
- 26 -
Pr( | )FC N j is given by Equations (11) and (12), and Pr( )FN j is obtained from Equation (14)
as a function of TDEi and M . Finally, T
DEi is obtained by finding the value that fulfills
1( )TDS DSp , with DSp according to Equation (19).
Validation
To validate the proposed model, we apply it to the simple 2-D frame structure shown in Figure
6. This structure is chosen because, despite its small number of elements, it captures some of
the characteristics of real structures. In particular, the structure exhibits redundancy with
respect to individual deterioration failures. The deterioration target reliability indices of the
structural elements are determined according to the proposed model. For validation, the
deterioration reliability of the system designed according to these target values is then
determined according to Equation (1), and is compared with the system deterioration target
reliability index. This comparison requires computing the reliability index of the system for all
n2 combinations of system deterioration states.
L
Elements 1-4: Elements 5-11: Top girder:
W18x130W18x76W36x150
1 2
3 4
5 6
7 8
9 10 11
7m
7m
3.5m
Figure 6. Structural system for model validation.
- 27 -
The considered structure is subjected to a random horizontal load, whose annual maximum L
has the Gumbel distribution with mean 351kNL and c.o.v. 0.35L . The material and
geometrical properties of the structural elements are modeled deterministically. The capacity of
the structure with respect to L is evaluated using non-linear FE (pushover) analysis. For the
intact structure, this capacity is assessed as 1461kN, which implies an annual reliability index
4.4DS
. It is assumed that deterioration can occur in structural elements 1-11, but not in the
top girders. Therefore, there are 112 2048 possible combinations of system deterioration
states and this number of pushover analyses are performed to evaluate Pr[ | ]iC for all
i.
Table 2 shows the resulting SEIi and corresponding ni values for the 11 elements, together with
the target reliability indices TDEi for different cases of T
DS and M , assuming ductile material
behaviour. The parameter k, which describes the number of equivalent Daniels systems, is
computed as k = 2.3 by Equation (17). (A non-integer value of k has no physical meaning, but
mathematically there is no difficulty in using such a value. The results obtained with a value of
k = 2.3 lie between results obtained with k = 2 and k = 3.)
Table 2. Resulting deterioration target reliability indices TDEi for the validation structure.
Assuming that at time t all elements have deterioration reliability indices exactly equal to their
target TDEi according to Table 2, the probability of each system deterioration state, Pr[ ]i ,
1, 2,..., 2048i , is computed. The true system deterioration reliability index DS of the
structural system in Figure 6 with the TDEi as given in Table 2 is then computed by Equation
(1). The results are summarized in Table 3. Also listed in the table in parentheses are true
system reliability indices obtained when using TDEi as determined by the current simplistic
method, which disregards the statistical dependence between deterioration failures.
Table 3. Resulting system deterioration reliability indices for the validation structural system (in parentheses: values obtained with the existing simplistic approach).
Target TDS DS
M = 0.0 M = 0.3 M = 0.6 3.7 3.5 (1.8) 3.4 (1.4) 3.5 (1.3) 4.2 4.1 (3.8) 4.0 (3.0) 4.0 (2.5)
As observed in Table 3, the proposed use of the idealized systems leads in all investigated
cases to a system deterioration reliability index that is close to but somewhat lower than the
system deterioration target reliability index. More striking, however, is the significant
improvement relative to the existing simplistic method. This is due to the approximate
accounting of the dependence between the deterioration failure events of the structural
elements by use of the equivalent Daniels systems.
Numerical investigation of influencing factors
The proposed model is applied to investigate the influence of the main input parameters. The
following base case is considered: 4.4SD
; 7.3TDS ; 20n ; 410iSEI for i = 1,…,n;
4.0M ; L is Lognormal distributed with 0.1L and c.o.v. 35.0L ; iR are Normal
distributed with c.o.v. 15.0iR ; all elements have ductile material behavior. These values of
- 29 -
SD and T
DS correspond to the case of a structure with large consequences of failure and with
normal cost of safety measures against overload failures and large cost of safety measures
against deterioration failures, see Table 1. Figure 7 presents the deterioration target reliability
index TDEi for the elements as a function of the system parameters.
βD
E i
T β
DE
iT
βD
E i
T β
DE
iT
βD
E i
T β
DE
iT
Single element importance measure SEIi System deterioration target reliability index βDS
Figure 7. Target reliability indices as a function of various influencing parameters.
The results in Figure 7 allow identifying the main influencing parameters. As expected, the
structural importance of the element, as expressed through the iSEI , is a key parameter (Figure
7a), as is the target reliability index for deterioration on the system level, TDS (Figure 7b). As
confirmed by the numerical investigations presented earlier, the statistical dependence among
deterioration safety margins has a strong influence on the system reliability (Figure 7c). The
resulting deterioration target reliability index for 4.0M is 4.3TDEi as opposed to
- 30 -
9.2TDEi for the case of no correlation 0.0M . This demonstrates that statistical
dependence among deterioration failure events of the elements must be considered when
determining the target reliability indices of redundant systems.
The reliability of the intact structure SD
has a moderate influence on TDEi (Figure 7d). T
DEi
increases with increasing SD
, which is due to the influence of SD
on the SEIi, Equation (7);
for fixed value of the SEIi, the probability of collapse given deterioration failure of element i
increases with increasing SD
. The influence of L , the c.o.v. of the annual maximum load on
the structure, is low (Figure 7e), which is fortunate, since this indicates that assumptions
regarding the overload failure mode of the structure are not critical when determining TDEi .
Figure 7f demonstrates that TDEi increases with increasing number of elements. This fact may
seem counter-intuitive but is due to the fact that the element structural importance is held
constant in the numerical investigation shown in Figure 7d. In reality, structures with more
elements tend to exhibit higher degrees of redundancy, thus having lower iSEI . To account for
this effect, Figure 8 presents TDEi for systems with varying degrees of redundancy. T
DEi is
shown as a function of iSEI , whereby the parameter describing the system size is held constant
as 5k . The number of elements is then computed as iknn , with in being a function of the
iSEI . As an example, for 310iSEI it follows that 2in and thus 10n , whereas for
510iSEI , 8in and 40n .
- 31 -
10-5 10-4 10-3 10-2 10-1 1.01.0
2.5
3.0
3.5
4.0
4.5
Single Element Importance Measure SEIi
1.5
2.0
Det
erio
ratio
n ta
rget
relia
bilit
y in
dex
β DEi
T
ρM = 0.6
ρM = 0.3
ρM = 0.0
Figure 8. Target reliability indices obtained according to the proposed Daniels system model for example systems with varying number of elements and corresponding SEIi.
Concluding remarks
As illustrated by the numerical examples in this paper, system effects, i.e., the joint effect of
several deterioration failures on the structural integrity, are relevant when determining target
reliability indices for deteriorating elements in redundant structural systems. However, a full
analysis of the system, which includes system reliability assessments for all combinations of
deterioration failures, is impractical for general structures. For this reason, highly simplified
system models have been used in the past to describe the effect of an element failure on the
integrity of the structure. These models do not represent the deterioration system effects
adequately and are not suitable for redundant structures. To account for the system effects in
determining acceptance criteria for individual deteriorating elements, this paper proposes using
idealized Daniels systems to represent the deteriorating elements in the structural system. This
is an idealization of the true system, which facilitates computation while capturing the overall
- 32 -
characteristics of the structural system, including its redundancy (load-sharing among
elements), and the influence of statistical dependence among deterioration failures on the
effective redundancy. Indicators for the structural importance of the system elements that have
been applied by previous approaches, such as the SEI, are used to define the characteristics of
the idealized Daniels systems. As demonstrated by the validation example, the proposed model
represents a significant improvement over current methods.
The proposed model is based on a number of idealizations and assumptions. In applying the
model, it must be checked whether these are justified, or whether the model must be extended.
Future research should be directed towards investigating applications for which these
assumptions do not hold. Two idealizations/assumptions of the model are deemed critical for a
number of applications: (a) the representation of deterioration by a two-state random variable,
which neglects that deterioration occurs gradually, and (b) disregard of progressive
deterioration failures. Concerning (a), future research efforts should be directed towards
identifying deterioration limit state functions which best represent the effect of deterioration on
the system reliability. It is noted that the current practice for defining deterioration failure is
often conservative, in particular for fatigue, where structural elements at failure still retain most
of their capacity. Concerning (b), progressive deterioration might be accounted for within the
existing model framework by assigning high correlation coefficients and an increased
probability of deterioration failure of the individual elements. Alternatively, the structural
elements that are jointly affected by the progressive deterioration mechanism might be
considered as a single (macro-)element in the system model.
- 33 -
Acknowledgements
This work was partially supported by the Swiss National Science Foundation (SNF) through grant PA002-111428.
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List of Figures
Figure 1: Idealized structural system under external load. Case a) brittle element behavior (original Daniels
system), b) ductile element behavior.
Figure 2: Failure probability of Daniels system as a function of the number of elements failed due to deterioration.
Figure 3: System deterioration reliability index as a function of number of elements (left chart), element
deterioration reliability index (middle chart), and correlation among deterioration limit states (right
chart).
Figure 4: The equivalent system for element i. All ni·k elements have independent identically distributed capacities
jiR , relative to overload failure. System failure occurs if any of the k subsystems fails.
Figure 5: Algorithm for establishing the equivalent systems.
Figure 6: Structural system for model validation.
Figure 7: Target reliability indices as a function of various influencing parameters.
Figure 8: Target reliability indices obtained according to the proposed Daniels system model for example systems
with varying number of elements and corresponding SEIi.
List of Tables
Table 1: Tentative target reliability indices T for ultimate limit states and one year reference period, as
recommended in JCSS (2006).
Table 2: Resulting deterioration target reliability indices for the validation structure.
Table 3: Resulting system deterioration reliability indices for the validation structural system (in parentheses: values obtained with the existing simplistic approach).
